Maple Lab for Calculus I
Lab 4
Limits, Infinity, and Asymptotes
Douglas Meade, Ronda Sanders, and Xian Wu
Department of Mathematics
Overview
The concept of a limit is an idea that all other calculus concepts are based on. Limits involving infinity
are closely related to asymptotes. While asymptotes for functions are sometimes easy to identify from
a graph, the actual definitions of asymptotes is in terms of limits. There are many different types of
asymptotes, and the two simplest are:
Asymptote
Horizontal
Vertical
Equation
y=L
x=a
Definition
lim f (x) = L or lim f (x) = L
x→∞
x→−∞
lim f (x) = ∞ (or −∞) or lim f (x) = ∞ (or −∞)
x→a+
x→a−
This lab is designed to provide experience finding asymptotes. Some Simple limits can easily be
evaluated by inspection. You will learn several ways to use Maple to help evaluate more difficult
limits.
Maple Essentials
Important Maple commands introduced in this lab:
Command
limit(f(x),x=a);
Description
evaluate lim f (x)
limit(f(x),x=a,right);
evaluate lim f (x)
limit((f(x)+1)/(x^2-1),x=1,right);
limit(f(x),x=a,left);
evaluate lim− f (x)
limit(sqrt(x^2+1)/(x+1),x=-1,left);
solve
solve equations
solve(x^2=8,x);solve(denom(f(x))=0,x);
x→a
x→a+
x→a
Example
limit(f(x),x=2);limit(x^2,x=infinity);
Related Text
§2.2, §2.3, and §2.6 of the Calculus Text and §4.1 of the Maple Text
Activities
1. Use Maple to find lim
h→0
(a)
(b)
(c)
(d)
f (x + h) − f (x)
for the following functions.
h
f (x) = x2
√
f (x) = x
f (x) = 1/x
f (x) = sin x
Note: You will soon learn that this limit defines the derivative of f (x).
2. Identify all horizontal and vertical asymptotes for as many of the functions on the back of this
page as possible.
Fall 2012
Created 8/28/2012
Maple Lab for Calculus I
Lab 4
General Directions
1. Define your function, say f (x), using f:=x-> and verify that it is entered correctly.
2. Look at the function f (x) and determine which values make the denominator zero. These values
will be the a’s that we need to check as possible vertical asymptotes. If you need help, you can
use the solve command as follows:
> solve(denom(f(x))=0,x);
3. To find the vertical asymptotes, you need to evaluate the one-sided limits at x = a. Enter the
following lines of code:
(a) limit(f(x), x=a, left);
(b) limit(f(x), x=a, right);
If either of these returns the value ∞ or −∞ then x = a is the equation of a vertical asymptote
of f (x).
4. To find the horizontal asymptotes of f (x), you need to find the limits at ∞ and −∞. Enter the
following lines of code:
(a) limit(f(x), x=infinity);
(b) limit(f(x), x=-infinity);
If either of these returns a value L 6= ±∞ then y = L is the equation of a horizontal asymptote
of f (x).
5. Plot the function on a standard window and make sure your answers are consistent with the
graph. Use the following line of code:
> plot(f(x), x=-10..10, y=-10..10, discont=true);
Functions
1. f (x) =
2. f (x) =
3. f (x) =
4. f (x) =
5. f (x) =
6. f (x) =
7. f (x) =
8. f (x) =
5+2x
x+1
3x+2
(x+2)2
√
(x2 −1) 4x2 +1
x3 −2x2 −x+2
sin x
x
x3 +3x2 −12x+4
x2
x3 +3x2 −12x+4
x3 −x2 −4x+4
√
x2 +1+2x
2x−3
√
9x2 +4−2
x+1
Assignment
For your assignment this week, complete the following and your TA may give additional problems.
Turn in a sheet of paper with your answers and attach your Maple worksheet as supporting work.
1. Answer each of the following questions. Explain all of your answers and use examples from today’s
lab where appropriate.
(a) Does every undefined value of f (x) lead to a vertical asymptote?
(b) Can the graph of a function cross the graph of its horizontal asymptotes?
(c) Can the graph of a function cross the graph of its vertical asymptotes?
(d) How many horizontal asymptotes can the graph of a function have?
2. Identify all horizontal and vertical asymptotes for the following functions.
(a) f (x) =
(b) h(x) =
Fall 2012
6x2 −10x−4
9x3 +27x2 −x−3
√
16x2 +3−8x
4x+1
Created 8/28/2012